Coloring of two-step graphs: open packing partitioning of graphs

Abstract

The two-step graphs are revisited by studying their chromatic numbers in this paper. We observe that the problem of coloring of two-step graphs is equivalent to the problem of vertex partitioning of graphs into open packing sets. With this remark in mind, it can be considered as the open version of the well-known 2-distance coloring problem as well as the dual version of total domatic problem.The minimum k for which the two-step graph N (G) of a graph G admits a proper coloring assigning k colors to the vertices is called the open packing partition number p o (G) of G, that is, p o (G) = χ N (G) . We give some sharp lower and upper bounds on this parameter as well as its exact value when dealing with some families of graphs like trees. Relations between p o and some well-know graph parameters have been investigated in this paper. We study this vertex partitioning in the Cartesian, direct and lexicographic products of graphs. In particular, we give an exact formula in the case of lexicographic product of any two graphs. The NP-hardness of the problem of computing this parameter is derived from the mentioned formula. Graphs G for which p o (G) equals the clique number of N (G) are also investigated.